Characterizations of inner product spaces. (English) Zbl 0617.46030

Amir, Dan

Operator Theory: Advances and Applications, Vol. 20. Basel - Boston - Stuttgart: Birkhäuser Verlag. VII, 200 p. DM 72.00 (1986).

This is an excellent, selfcontained, up-to-date, exhaustive survey on various isometric characterizations of inner product spaces. In Chapter 2 on 2-dimensional characterizations the author discusses among others James and Birkhoff orthogonality characterizations as well as norm derivatives and best approximation characterizations.
In Chapter 3 on 3-dimensional characterizations the characterizations included range from classical Kakutani’s result, through metric projections characterizations to Chebyshev centers and proximinal maximal subspaces, to name just a few topics covered in the book.
The book is well written, well organized. It bears a seal of expertize of the author who is a renowned specialist in the theory of normed spaces.
The book will certainly be a useful source for researchers both in this area as well as in many related areas of functional analysis, geometry and approximation theory.

Reviewer: V.Zizler

MSC:

46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
46B20	Geometry and structure of normed linear spaces
46-02	Research exposition (monographs, survey articles) pertaining to functional analysis

Keywords:

isometric characterizations of inner product spaces; James and Birkhoff orthogonality; norm derivatives; best approximation; Chebyshev centers; proximinal maximal subspaces